Rend. Sem. Mat. Univ. Pol. Torino Vol. 58, 1 2000
Geom., Cont. and Micros., I
M. Epstein
∗
ARE CONTINUOUS DISTRIBUTIONS OF INHOMOGENEITIES IN LIQUID CRYSTALS POSSIBLE?
Abstract. Within a theory of liquid-crystals-like materials based on a generalized Cosserat-type formulation, it is shown that continuous distributions of inhomo-
geneities may exist at the microstructural level.
1. Introduction
In the conventional theories of liquid crystals, the free-energy density is assumed to be a function of a spatial vector field and its spatial gradient. Starting from the pioneering work of Frank
[6], various improvements were proposed by Leslie [9] and by Ericksen [4] [5]. A different point of view was advocated by Lee and Eringen [7] [8], as early as 1972, when considering a
liquid criystal within the framework of the theory of materials with internal structure. The main difference between these points of view is that the second approach emphasizes the dependence
of the constitutive equations on the mappings between vectors or tensor fields, rather than on their values alone. This mapping-dependence is essential not only for sustaining continuous
distributions of inhomogeneities, but also, as shown by Maugin and Trimarco [10], for the proper setting of a definition of Eshelby stresses. The general connection between these two aspects of
material behaviour is described in [3].
2. The generalized Cosserat medium
A generalized Cosserat body GCB consists of the frame bundle of an ordinary body B. In other words, a GCB is a body plus the collection of all its local frames at each point. Denoting by X
I
I = 1, 2, 3 and x
i
i = 1, 2, 3 Cartesian coordinate systems for the body B and for physical
space, respectively, a configuration of a GCB consists of the twelve independent functions: x
i
= x
i
X
J
H
i I
= H
i I
X
J
where H
i I
represents the mapping of the frames attached at point X
J
. It is important to stress that the ordinary deformation gradient F
i I
=
∂ x
i
∂ X
I
and the mapping H
i I
are of the same nature, but represent two independent vector-dragging mechanisms.
A GCB is hyperelastic of the first grade if its material response can be completely charac- terized by a single scalar “strain-energy” function:
W = W F
i I
, H
i I
, H
i I, J
; X
K
∗
Partially supported by the Natural Sciences and Engineering Research Council of Canada, and DGI- CYT Spain Project PB97-1257.
93
94 M. Epstein
where comma subscripts denote partial derivatives. Under a change of reference configuration of the form
Y
A
= Y
A
X
J
H
A I
= H
A I
X
J
where the indices A, B, C are used for the new reference, the energy function changes to: W
= W
′
F
i A
, H
i A
, H
i A,B
; Y
C
= W F
i A
F
A I
, H
i A
H
A I
, H
i A,B
F
B J
H
A I
+ H
i A
H
A I, J
; X
K
Y
C
1 Notice the special form of the composition law for the derivatives of H
i I
. Generalizing Noll’s idea of uniformity [11], by taking into account the composition laws in
Equation 1, one can show [1] [2] that in terms of an archetypal energy function W
c
= W
c
F
i α
, H
i α
, H
i αβ
where Greek indices are used for the archetype, a GCB is uniform namely, it is made of “the same material” at all points if there exist three uniformity fields of tensors P
I α
X
J
, Q
I α
X
J
and R
I αβ
X
J
such that the equation W F
i I
, H
i I
, H
i I, J
; X
K
= W
c
F
i I
P
I α
, H
i I
Q
I α
, H
i I, J
P
J β
Q
I α
+ H
i I
R
I αβ
is satisfied identically for all non-singular F
i I
and H
i I
and for all H
i I, J
. Homogeneity global or local follows if, and only if, there exists a global or local reference configuration such that
these fields become trivial.
3. The liquid-crystal-like model
Kategori